| L(s) = 1 | − 2·2-s + 12·4-s + 96·7-s − 104·8-s + 404·9-s − 192·14-s − 592·16-s + 200·17-s − 808·18-s + 2.33e3·23-s + 7.02e3·25-s + 1.15e3·28-s − 1.29e4·31-s + 1.63e3·32-s − 400·34-s + 4.84e3·36-s − 4.56e3·41-s − 4.67e3·46-s − 5.47e4·47-s − 2.40e4·49-s − 1.40e4·50-s − 9.98e3·56-s + 2.58e4·62-s + 3.87e4·63-s − 1.36e4·64-s + 2.40e3·68-s + 2.06e5·71-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 3/8·4-s + 0.740·7-s − 0.574·8-s + 1.66·9-s − 0.261·14-s − 0.578·16-s + 0.167·17-s − 0.587·18-s + 0.920·23-s + 2.24·25-s + 0.277·28-s − 2.41·31-s + 0.281·32-s − 0.0593·34-s + 0.623·36-s − 0.424·41-s − 0.325·46-s − 3.61·47-s − 1.43·49-s − 0.795·50-s − 0.425·56-s + 0.854·62-s + 1.23·63-s − 0.416·64-s + 0.0629·68-s + 4.86·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.311995608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.311995608\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $D_{4}$ | \( 1 + p T - p^{3} T^{2} + p^{6} T^{3} + p^{10} T^{4} \) |
| good | 3 | $C_2^2 \wr C_2$ | \( 1 - 404 T^{2} + 9350 p^{2} T^{4} - 404 p^{10} T^{6} + p^{20} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 - 7028 T^{2} + 24404246 T^{4} - 7028 p^{10} T^{6} + p^{20} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 48 T + 15502 T^{2} - 48 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 296436 T^{2} + 49128544726 T^{4} - 296436 p^{10} T^{6} + p^{20} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 894228 T^{2} + 396323515894 T^{4} - 894228 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 100 T + 2767462 T^{2} - 100 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 6794580 T^{2} + 21506967947254 T^{4} - 6794580 p^{10} T^{6} + p^{20} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 1168 T + 10055470 T^{2} - 1168 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 31255380 T^{2} + 976386653995702 T^{4} - 31255380 p^{10} T^{6} + p^{20} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 6464 T + 65013054 T^{2} + 6464 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 241262580 T^{2} + 24149916431784598 T^{4} - 241262580 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 2284 T + 146603254 T^{2} + 2284 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 10845276 p T^{2} + 96250708269010006 T^{4} - 10845276 p^{11} T^{6} + p^{20} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 27360 T + 591338206 T^{2} + 27360 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 1039152180 T^{2} + 595616955270391126 T^{4} - 1039152180 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 1537424180 T^{2} + 1399694789142612374 T^{4} - 1537424180 p^{10} T^{6} + p^{20} T^{8} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 + 741098540 T^{2} + 1478044222094100534 T^{4} + 741098540 p^{10} T^{6} + p^{20} T^{8} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 1366835860 T^{2} + 3274116308996825526 T^{4} - 1366835860 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 103344 T + 6217736974 T^{2} - 103344 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 19988 T + 2541602870 T^{2} - 19988 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 123936 T + 9855929374 T^{2} + 123936 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 10047855188 T^{2} + 55381071937674414326 T^{4} - 10047855188 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 42316 T + 4292401174 T^{2} + 42316 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 49788 T + 16391371462 T^{2} + 49788 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.98056003575441920927910929848, −15.46159282788681824118615799503, −15.32388784117557855982143589838, −14.62218622970180512937539015126, −14.52953575703698600552459463057, −14.10746190163334342025484236232, −13.11881435984734104265713163009, −13.07464035345282286936694309402, −12.55298805903256300934222531860, −12.40471134696526915104462435550, −11.21040594373596566035310849962, −11.16966677750468474291321996939, −11.11460151679582914023082211631, −9.921819300493475266502780784502, −9.912456974173941537052175492134, −9.233917740109243373165274818869, −8.557130676955492868920538131225, −8.205562942966769425332630197734, −7.12561003504052907701598768524, −7.10216428735290904435596455564, −6.41092439415696637848192265195, −5.11224965310140540215494532693, −4.71880582775102599709882316828, −3.37396394190873989545603734583, −1.63932226537607778906786474852,
1.63932226537607778906786474852, 3.37396394190873989545603734583, 4.71880582775102599709882316828, 5.11224965310140540215494532693, 6.41092439415696637848192265195, 7.10216428735290904435596455564, 7.12561003504052907701598768524, 8.205562942966769425332630197734, 8.557130676955492868920538131225, 9.233917740109243373165274818869, 9.912456974173941537052175492134, 9.921819300493475266502780784502, 11.11460151679582914023082211631, 11.16966677750468474291321996939, 11.21040594373596566035310849962, 12.40471134696526915104462435550, 12.55298805903256300934222531860, 13.07464035345282286936694309402, 13.11881435984734104265713163009, 14.10746190163334342025484236232, 14.52953575703698600552459463057, 14.62218622970180512937539015126, 15.32388784117557855982143589838, 15.46159282788681824118615799503, 15.98056003575441920927910929848
